2. Introduction
● A Fuzzy Inference System (FIS) is a way of mapping
an input space to an output space using fuzzy logic
● FIS uses a collection of fuzzy membership functions
and rules, instead of Boolean logic, to reason about
data.
● The rules in FIS (sometimes may be called as fuzzy
expert system) are fuzzy production rules of the form:
− if p then q, where p and q are fuzzy statements.
● For example, in a fuzzy rule
− if x is low and y is high then z is medium.
− Here x is low; y is high; z is medium are fuzzy statements; x
and y are input variables; z is an output variable, low, high,
and medium are fuzzy sets.
3. Cont…
● The antecedent describes to what degree the rule
applies, while the conclusion assigns a fuzzy function
to each of one or more output variables.
● Most tools for working with fuzzy expert systems allow
more than one conclusion per rule.
● The set of rules in a fuzzy expert system is known as
knowledge base.
● The functional operations in fuzzy expert system
proceed in the following steps.
− Fuzzification
− Fuzzy Inferencing (apply implication method)
− Aggregation of all outputs
− Defuzzification
5. Fuzzification
● In the process of fuzzification, membership functions
defined on input variables are applied to their actual
values so that the degree of truth for each rule
premise can be determined.
● Fuzzy statements in the antecedent are resolved to a
degree of membership between 0 and 1.
− If there is only one part to the antecedent, then this is the
degree of support for the rule.
− If there are multiple parts to the antecedent, apply fuzzy logic
operators and resolve the antecedent to a single number
between 0 and 1.
● Antecedent may be joined by OR; AND operators.
− For OR -- max
− For AND -- min
6. Fuzzy Inferencing
● In the process of inference
− Truth value for the premise of each rule is computed and
applied to the conclusion part of each rule.
− This results in one fuzzy set to be assigned to each output
variable for each rule.
● The use of degree of support for the entire rule is to
shape the output fuzzy set.
● The consequent of a fuzzy rule assigns an entire fuzzy
set to the output.
● If the antecedent is only partially true, (i.e., is assigned
a value less than 1), then the output fuzzy set is
truncated according to the implication method.
7. Cont...
● If the consequent of a rule has multiple parts, then all
consequents are affected equally by the result of the
antecedent.
● The consequent specifies a fuzzy set to be assigned
to the output.
● The implication function then modifies that fuzzy set
to the degree specified by the antecedent.
● The following functions are used in inference rules.
● min or prod are commonly used as inference rules.
− min: truncates the consequent's membership function
− prod: scales it.
8. Aggregation of all outputs
● It is the process where the outputs of each rule are
combined into a single fuzzy set.
● The input of the aggregation process is the list of
truncated output functions returned by the implication
process for each rule.
● The output of the aggregation process is one fuzzy set
for each output variable.
− Here, all fuzzy sets assigned to each output variable are
combined together to form a single fuzzy set for each output
variable using a fuzzy aggregation operator.
● Some of the most commonly used aggregation
operators are
− the maximum : point-wise maximum over all of the fuzzy sets
− the sum : (point-wise sum over all of the fuzzy
− the probabilistic sum.
9. Defuzzification
● In Defuzzificztion, the fuzzy output set is converted to
a crisp number.
● Some commonly used techniques are the centroid and
maximum methods.
− In the centroid method, the crisp value of the output variable is
computed by finding the variable value of the centre of gravity
of the membership function for the fuzzy value.
− In the maximum method, one of the variable values at which
the fuzzy set has its maximum truth value is chosen as the
crisp value for the output variable.
● Some other methods for defuzzification are:
− bisector, middle of maximum (the average of the maximum
value of the output set), largest of maximum, and smallest of
maximum, etc.
10. Generic Method
● Main steps are
− Evaluate the antecedent for each rule
− Obtain each rule's conclusion
− Aggregate conclusions
− Defuzzification
● We will explain these steps using an example of
Tipping Problem
● Two inputs : Quality of food and Service at a restaurant
rated at scale from 0-10
● One output: Amount of tip to be given
● Tip should reflect the quality of the food and service.
● The tip might be in the range 5-15% of total bill paid.
11. Rules for Tipping
● Let us consider the following three rules
− If service is poor or food is bad, then tip is cheap
− If service is good, then tip is average
− If service is excellent or food is delicious, then tip is
generous
● Input variables
− Service : represented by poor, good, excellent
− Food : represented by bad, delicious
● Output Variable:
− Tip : represented by cheap, average, generous
18. Introduction
● MATLAB fuzzy logic toolbox provides facility for the
development of fuzzy-logic systems using
− graphical user interface (GUI) tools
− command line functionality
● There are five primary GUI tools
− Fuzzy Inference System (FIS) Editor
− Membership Function Editor
− Rule Editor
− Rule Viewer
− Surface Viewer
21. Membership Function Editor
Display & edit
values of current
variable
Select & edit
attributes of
membership
function
Name & edit
parameters of
membership
function
